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Mirrors > Home > MPE Home > Th. List > domentr | Structured version Visualization version GIF version |
Description: Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.) |
Ref | Expression |
---|---|
domentr | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endom 8655 | . 2 ⊢ (𝐵 ≈ 𝐶 → 𝐵 ≼ 𝐶) | |
2 | domtr 8681 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan2 596 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 class class class wbr 5053 ≈ cen 8623 ≼ cdom 8624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-f1o 6387 df-en 8627 df-dom 8628 |
This theorem is referenced by: domdifsn 8728 xpdom1g 8742 domunsncan 8745 sdomdomtr 8779 domen2 8789 mapdom2 8817 php 8830 unxpdom2 8886 sucxpdom 8887 xpfir 8897 fodomfi 8949 cardsdomelir 9589 infxpenlem 9627 xpct 9630 infpwfien 9676 inffien 9677 mappwen 9726 iunfictbso 9728 djuxpdom 9799 cdainflem 9801 djuinf 9802 djulepw 9806 ficardun2 9816 ficardun2OLD 9817 unctb 9819 infdjuabs 9820 infunabs 9821 infdju 9822 infdif 9823 infxpdom 9825 pwdjudom 9830 infmap2 9832 fictb 9859 cfslb 9880 fin1a2lem11 10024 fnct 10151 unirnfdomd 10181 iunctb 10188 alephreg 10196 cfpwsdom 10198 gchdomtri 10243 canthp1lem1 10266 pwfseqlem5 10277 pwxpndom 10280 gchdjuidm 10282 gchxpidm 10283 gchpwdom 10284 gchhar 10293 inttsk 10388 inar1 10389 tskcard 10395 znnen 15773 qnnen 15774 rpnnen 15788 rexpen 15789 aleph1irr 15807 cygctb 19277 1stcfb 22342 2ndcredom 22347 2ndcctbss 22352 hauspwdom 22398 tx2ndc 22548 met1stc 23419 met2ndci 23420 re2ndc 23698 opnreen 23728 ovolctb2 24389 ovolfi 24391 uniiccdif 24475 dyadmbl 24497 opnmblALT 24500 vitali 24510 mbfimaopnlem 24552 mbfsup 24561 aannenlem3 25223 dmvlsiga 31809 sigapildsys 31842 omssubadd 31979 carsgclctunlem3 31999 finminlem 34244 phpreu 35498 lindsdom 35508 mblfinlem1 35551 pellexlem4 40357 pellexlem5 40358 pr2dom 40819 tr3dom 40820 nnfoctb 42268 ioonct 42750 subsaliuncl 43572 caragenunicl 43737 aacllem 46176 |
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